The article considers the problem of constructing a -periodic solution of a quasilinear second-order integro-differential equation. Using the Green’s function of bounded solutions on the number line, the integro-differential equation is reduced to an integral equation. A -periodic solution to the integral equation is found using the projection-iteration method. A -periodic solution is sought as the limit of successive -periodic functions representable as a Fourier series. An estimate of the error of the difference between the exact and approximate solutions is obtained.

periodic solutions, quasilinear second-order integro-differential equations, Green’s function, integral equations on the number axis, exact and approximate solutions, method of successive approximations